Worksheet: Approximating Radicals

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In this worksheet, we will practice approximating the nth root of a number and then using it to solve real-world problems.

Q1:

Which of the following numbers is closest to 5?

  • A√106
  • B√8
  • C√24
  • D√94
  • E√120

Q2:

The formula for the area of a square is 𝐴=𝑠, where 𝑠 is the side length. Estimate the side length of a square whose area is 74 square inches.

Q3:

By estimating the values of the square roots to the nearest whole number, order the following from least to greatest: √38, 7, √88, 4.

  • A7,√88,√38,4
  • B4,7,√38,√88
  • C4,√38,7,√88
  • D√38,4,7,√88
  • E√88,7,√38,4

Q4:

Estimate √38 to the nearest whole number.

Q5:

Estimate 43310 to the nearest whole number.

Q6:

Estimate the solution of the following equation to the nearest integer: 𝑑=68.

  • A𝑑=34 or 𝑑=−34
  • B𝑑=68 or 𝑑=−68
  • C𝑑=3 or 𝑑=−3
  • D𝑑=8 or 𝑑=−8

Q7:

What is the largest integer 𝑎 such that both 1𝑎 and 𝑎√463 are between 0 and 1?

Q8:

Given that 𝑥∈ℤ and 𝑥<√15<𝑥+1, find the value of 𝑥.

Q9:

Without using a calculator, find two consecutive integers that √13+√6 lies between.

  • A5, 6
  • B6, 7
  • C7, 8
  • D4, 5

Q10:

Arrange the following numbers in descending order: √33,−√22,√41,3,0,√−125.

  • A3,√33,√41,0,−√22,√−125
  • B√41,√33,3,0,−√22,√−125
  • C√41,√33,3,0,√−125,−√22
  • D3,√33,√41,0,√−125,−√22

Q11:

Arrange the following numbers in ascending order: √129,−√91,√49,5,0,√−125.

  • A−√91,√−125,0,√129,√49,5
  • B−√91,√−125,0,5,√49,√129
  • C√−125,−√91,0,√129,√49,5
  • D√−125,−√91,0,5,√49,√129
  • E√129,√49,5,0,√−125,−√91

Q12:

Arrange the following numbers in descending order: √83,1,−√89,√97.

  • A√97,1,√83,−√89
  • B−√89,1,√83,√97
  • C√97,√83,1,−√89
  • D√83,√97,1,−√89

Q13:

Find an approximated value of √57−40 without using a calculator.

  • A−32.4 or −32.5
  • B55 or 55.1
  • C63.3 or 63.4
  • D−47.5 or −47.6
  • E47.5 or 47.6

Q14:

Find the two consecutive one-decimal-place numbers that √151 lies between.

  • A12.3, 12.4
  • B12.1, 12.2
  • C12.2, 12.3
  • D12.4, 12.5

Q15:

The positions of the numbers √120, √102, and √111 have been identified on the number line. By considering their size, decide which number is represented by 𝑎.

  • A√120
  • B√111
  • C√102

Q16:

The positions of the numbers √24, √17, √22, and √19 have been highlighted on the number line. Which arrow represents the position of √19?

  • Aa
  • Bc
  • Cd
  • Db

Q17:

The positions of the numbers √24, √17, √22, and √19 have been identified on the number line. Which arrow represents the position of √22?

  • Ab
  • Ba
  • Cd
  • Dc

Q18:

Michael is trying to find which two whole numbers lie on either side of √51. He decides that it will be helpful to use what he has been taught about square numbers.

What is the greatest square number below 51?

What is the smallest square number above 51?

Hence, determine the two consecutive whole numbers that √51 lies between.

  • A6 and 9
  • B8 and 9
  • C7 and 9
  • D7 and 8
  • E6 and 8

Q19:

Daniel wants to find an approximation for √2 accurate to two decimal places. He sets up the equation 𝑥=2 and then starts using trial and improvement to find the solution as seen in the table.

𝑥𝑥=2big/small
11=1s
22=4b
1.51.5=2.25b
1.41.4=1.96s
1.41
1.42
1.415

Determine the value of 1.41 giving your answer accurate to three decimal places.

Determine the value of 1.42 giving your answer accurate to three decimal places.

Determine the value of 1.415 giving your answer accurate to three decimal places.

Hence, what is the value of √2 accurate to two decimal places?

Q20:

The geometric mean of two numbers 𝑎 and 𝑏 can be found by evaluating √𝑎⋅𝑏. Estimate the geometric mean of 2 and 9 to the nearest whole number.

Q21:

Without using a calculator, determine which of √54 or 11 is greater.

  • A11
  • B√54

Q22:

Given that a whole number is squared and that the result is between 875 and 974, determine between which of the following is the whole number.

  • A29 and 31
  • B27 and 29
  • C33 and 35
  • D25 and 27
  • E31 and 33

Q23:

By considering square numbers, determine an approximation for √83 to the nearest whole number.

Q24:

The radius 𝑟 of a sphere is given by the formula 𝑟=3𝑉4𝜋, where 𝑉 is the sphere’s volume. Determine the difference in radius between a sphere with volume 7,776𝜋 and a sphere with volume 18,432𝜋.

Q25:

Which of the integers below is the closest to √−353?

  • A−17
  • B−6
  • C−18
  • D−7

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